Self-Improvement Under Diagonal Constraints: No Universal Self-Upgrade Certifier, Stratified Improvement, and Evolution as an Attractor Architecture Paper 32 of the NEMS Suite

Papers 26–31 developed an abstract toolkit for self-reference (Paper 26), closure audits (Paper 27), reflection as a resource (Paper 28), strength-ordered barrier schemas (Paper 29), self-trust incompleteness (Paper 30), and social verification as a theorem-grade protocol with diversity necessity (Paper 31). The present paper applies that toolkit to self-improvement. We formalize proposed upgrades as certificates and "this upgrade is good/safe" as a claim predicate. We prove the Self-Improvement Barrier Theorem: under anti-decider closure and diagonal capability (availability of the fixed-point premise for the relevant transformer class), no agent has a universal total internal certifier for any nontrivial extensional upgrade predicate. We then prove a stratified improvement principle: below diagonal closure (when is not supplied), total internal verifiers may exist for restricted upgrade classes; but at or above diagonal closure, universal self-certification is impossible. Next, importing Paper 31, we show society improves improvement: role-separated external auditing protocols can strictly expand the verified upgrade coverage relative to any individual verifier, and diversity is necessary for strict improvement. These results are formal statements about coverage, admissible aggregation, and diagonal-capable verification architectures; broader institutional analogies are interpretations, not the theorem statements themselves. Finally, we interpret these theorem-backed constraints as a general design pressure toward population-based external validation—an evolution-like architecture in which diversity generates variants and the environment supplies verification. The development is mechanized in Lean 4 as the SelfImprovement library in nems-lean, with zero sorry and no custom axioms. This overview presents the core NEMS theorem engine and selected applications; stronger domain-specific derivation and ontological synthesis claims belong to separate release surfaces with their own premise bundles and formal artifacts. Trust boundary. The self-improvement barrier is indexed by anti-decider closure and diagonal capability; society-improvement results import Paper 31 protocol premises; evolutionary interpretations are informal. Mechanization is nems-lean. See.